Exponential Growth in Flat Spacetime
Consider an n+1-dimensional flat spacetime full of n-dimensional bacteria that are:
1. Exponentially growing The rate at which a clump of bacteria increases its volume is proportional to its volume. That is, for a clump of volume V:
dV/dτ = rV Where r is some constant and τ is time as measured by the clump of bacteria. 2. Incompressible All the bacteria are moving with some position-dependent speed β(x) to make room for their neighbors to grow. So, if a stationary observer measures a clump of bacteria to have diameter Δx at time t:
β(x + Δx) - β(x) = dΔx/dt
The volume of a clump of bacteria is proportional to the nth power of its diameter. That is, for some constant k, if the clump of bacteria measures its own diameter to be Δs:
V = k(Δs)n
Then using the chain rule and the formulae for length contraction and time dilation:
dΔx/dt = dΔx/dΔs dΔs/dV dV/dτ dτ/dt = 1/γ 1/nk(Δs)n-1 rV 1/γ
Simplifying the above, we find:
β(x + Δx) - β(x) = rΔx/γn = rΔx/n √(1 - β2)
In the limit as Δx goes to zero, this becomes a differential equation:
dβ/dx = r/n √(1 - β2)
With the initial conditions β = 0, x = 0, (observed by the bacterium who's lucky enough to be stationary) the solution is:
β = sin (rx/n)
Interpretation
For exponential growth to be sustained, bacteria must be moving at the speed of light (β = 1) when they reach x = nπ/2r. Then if the speed of light is unreachable, there's no way to sustain exponential growth! A corollary is that the quiverfulls need to put a cork in it, because even going into space and colonizing Mars won't save us from the growth limit.
I will freely admit that I completely ignored gravitational time dilation, spacetime curvature, dark energy, the energy necessary to accelerate bacteria outward like that, and the horrendous pressure on the poor bacterium at x = 0. The reader is cordially invited to take any of these factors into consideration and see where that leads.